Know-How Exchange : Just for Fun

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Dear Adam,

Your question appears to be a direct quote from H. C. Tijms' book "Understanding Probability: Chance Rules in Everyday Life"

https://tinyurl.com/krp7az

So I'm wondering, is this some kind of homework assignment?

As a matter of fact I do not know where it comes from. No, it's not my homework as at my age I do not do it anymore. Just a friend of mine sent it to me asking whether as he it seems to him the best strategy is to draw 3 marbles. I'm not quite sure, that's why I posted teh question. Simple as that.
If you do not like it or do not want to give you answer, that's ok by me.

Adam

Adam,
If I guess wrong, what have I lost? So...pull 3

Michael
You lose nothing as you invest your time and hope only.

Gary
Thanks for your insight. My answer to my friend's question was quite similar. Take a gamble. Even if you start with 10,000 you and have ar. 70 odd percent chance of winning 8500 instead of 50/50 chance of winning 10,000 it makes sense.

Adam

Dear Adam

There’s something wrong in the wording of the question:, “The vase contains a very large number of marbles. You must guess whether the vase has twice as many white ones as red ones.”

What – guess whether the vase has twice as many white ones as red ones as against any other combination from 1 white one and n-1 red ones where n is the total number or marbles or is the question of the form that there are n/3 red and 2n/3 white where n/3 is an integer and the only other alternative is that the vase contains 2n/3 red and n/3 white.

The 1, 2 or 3 marble test is to assess the likelihood of the n/3 fraction being white or the n/3 fraction being red.

Could you clarify?

Also you don’t have to use Bayes. Bayes is most useful when there is an assumed or known association between the observations of a variable and the likelihood of a particular outcome for the variable. For example if an animal can enter a chamber out of either of two holes, behind one of which might or might not be it’s nest, the odds of it using one over the other is determined by connected observations such as spoil haps or tracks. If the nest is actually behind one of the holes than the odds of it using that hole move from 50% of outings to nearly 100%. Observing the thing three times doesn’t tell you anything unless you are allowed to infer the relationship – then Bayes comes into play.

Regardless, if you’ll clarify the question, I’ll give you my shot at the answer and the workings which Bayes and with normal distributions.

Actually for normal distributions to apply, you have to define “Many” If by many you mean a number large enough to make (Many) almost equal to (Many-1) then that’s one thing. If you mean that you should differentiate between (many-1), (many-2) and (many-3), then that’s a different ball game. The first has a solution in the calculus, the second doesn’t!

Best wishes

Steve

it depends. Are you married?

If you are not married, then you should use Bayes law, which I know nothing about, and make some sort of rational decision.

But if you are married, you should not buy any balls. You should just guess. If you win, your wife will spend the money anyway. If you buy any balls, and you lose, you will hear about it for the rest of your life.

Michael

The problem with that, if I’ve interpreted between the lines is that if many = a really, really large number, the pulling 3 lowers the prize without altering the odds in any significant manner.

Telemoxie’s idea of basing strategy on whether you are married or not holds more water!

Steve

Michael yes, you are right. I double checked with my friend and it should read" twice as many red marbles as white marbles or twice as many white marbles as red marbles. Sorry for lack of my attention.
As to many marbles in the vase I take it be so many as to equivalent to drawing with replacement.

Pls forgive me Steve I meant to address you not Michael.
That goddamn iPhone once again and my clumsiness. Once again sorry for my mistake.

Adam

Thks Juliet, I enjoyed you posting a great deal. As a matter of fact it's a fun forum isn't it?
Using probabilities strategy no.3 is the best, but not by a big margin. I can win $8,500 with a probability ar.74% giving expected value of $ 6,295. However most game players( ar.70%) in real life-game choose strategy 1 of drawing 1 ball only with which they have a 2/3 probability of winning $9,250 i.e. $6,167. Does it mean they do not follow the rule of maximizing expected value if it involves too much trouble and is time-consuming and are happy with good enough strategies to satisfice instead of optimze?

Adam

you say that one possibility is that there are twice as many red marbles as white marbles, and you say that each possibility is equally likely. But what is the other possibility?

Is the other possibility that there is an even number of red and white marbles? Or is the other possibility that there are twice as many white marbles as red marbles?or is the other possibility that there is any imaginable percentage of white balls and red balls?

It makes a difference...

A good question. The vase has twice as many white marbles as red marbles or twice as many red marbles as white marbles.

Adam

Oh, I nearly forgot. There is a quantum solution where the eigenvalues of the wave equation defining red or white marbles allows a marble to be both red and white at the same time. It assumes redness or whiteness only when you observe it. (Eigenvector uncertainty at the level of the Plank constant)

Assuming that you only calculate the answer and never look, you can always win the largest prize by entangling the answer with the question.

If this place had somewhere to do the maths notation I’d show you the arguments, but in text, it would look like about a mile of computer printout.

Or toilet paper


Steve

Steven you got perplexed with your maths skills. Yours was the most comprehensive answer from computational point of view. I also appreciate Gary's reply which seems to epitomize fast and frugal heuristics people use to get the most of it while sparing extra effort and trouble.
My sincere thanks to all guys who contributed their answers without prejudice.

Adam

Thanks Adam

I should have added - the quantum solution only applies to very small marbles.

Steve